An Estimate of The Probability of Alignments Existing By Chance
Statistically, finding alignments on a landscape gets progressively easier as the geographic area to be considered increases. One way of understanding this phenomenon is to see that the increase in the number of possible combinations of sets of points in that area overwhelms the decrease in the probability that any given set of points in that area line up.
The number of alignments found is very sensitive to the allowed width w, increasing approximately proportionately to wk-2, where k is the number of points in an alignment.
For those interested in the mathematics, the following is a very approximate order-of-magnitude estimate of the likelihood of alignments, assuming a plane covered with uniformly distributed "significant" points.
Consider a set of n points in a compact area with approximate diameter d and area approximately d². Consider a valid line to be one where every point is within distance w/2 of the line (that is, lies on a track of width w, where w << d).
Consider all the unordered sets of k points from the n points, of which there are:
What is the probability that any given set of points is collinear in this way? Let us very roughly consider the line between the "leftmost" and "rightmost" two points of the k selected points (for some arbitrary left/right axis: we can choose top and bottom for the exceptional vertical case). These two points are by definition on this line. For each of the remaining k-2 points, the probability that the point is "near enough" to the line is roughly w/d, which can be seen by considering the ratio of the area of the line tolerance zone (roughly wd) and the overall area (roughly d²).
So, the expected number of k-point alignments, by this definition, is very roughly:
For n >> k this is approximately:
Now assume that area is equal to, and say there is a density α of points such that .
Then we have the expected number of lines equal to:
and an area density of k-point lines of:
Gathering the terms in k we have an areal density of k-point lines of:
Thus, contrary to intuition, the number of k-point lines expected from random chance increases much more than linearly with the size of the area considered.
Read more about this topic: Alignments Of Random Points
Famous quotes containing the words estimate, probability, existing and/or chance:
“It is, indeed, at home that every man must be known by those who would make a just estimate either of his virtue or felicity; for smiles and embroidery are alike occasional, and the mind is often dressed for show in painted honour, and fictitious benevolence.”
—Samuel Johnson (1709–1784)
“Crushed to earth and rising again is an author’s gymnastic. Once he fails to struggle to his feet and grab his pen, he will contemplate a fact he should never permit himself to face: that in all probability books have been written, are being written, will be written, better than anything he has done, is doing, or will do.”
—Fannie Hurst (1889–1968)
“The primary function of myth is to validate an existing social order. Myth enshrines conservative social values, raising tradition on a pedestal. It expresses and confirms, rather than explains or questions, the sources of cultural attitudes and values.... Because myth anchors the present in the past it is a sociological charter for a future society which is an exact replica of the present one.”
—Ann Oakley (b. 1944)
“There is a history in all men’s lives,
Figuring the natures of the times deceased,
The which observed, a man may prophesy,
With a near aim, of the main chance of things
As yet not come to life.”
—William Shakespeare (1564–1616)